1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
|
.. _continuous-gamma:
Gamma Distribution
==================
The standard form for the gamma distribution is :math:`\left(\alpha>0\right)` valid for :math:`x\geq0` .
.. math::
:nowrap:
\begin{eqnarray*} f\left(x;\alpha\right) & = & \frac{1}{\Gamma\left(\alpha\right)}x^{\alpha-1}e^{-x}\\
F\left(x;\alpha\right) & = & \frac{\gamma\left(\alpha,x\right)}{\Gamma(\alpha)}\\
G\left(q;\alpha\right) & = & \gamma^{-1}\left(\alpha,q\Gamma(\alpha)\right)\end{eqnarray*}
where :math:`\gamma` is the lower incomplete gamma function, :math:`\gamma\left(s, x\right) = \int_0^x t^{s-1} e^{-t} dt`.
.. math::
M\left(t\right)=\frac{1}{\left(1-t\right)^{\alpha}}
.. math::
:nowrap:
\begin{eqnarray*} \mu & = & \alpha\\
\mu_{2} & = & \alpha\\
\gamma_{1} & = & \frac{2}{\sqrt{\alpha}}\\
\gamma_{2} & = & \frac{6}{\alpha}\\
m_{d} & = & \alpha-1\end{eqnarray*}
.. math::
h\left[X\right]=\Psi\left(a\right)\left[1-a\right]+a+\log\Gamma\left(a\right)
where
.. math::
\Psi\left(a\right)=\frac{\Gamma^{\prime}\left(a\right)}{\Gamma\left(a\right)}.
Implementation: `scipy.stats.gamma`
|
